(1) Field of the Invention
This invention relates to the field of structural properties, and in particular to the determination of the complex flexural wavenumber, corresponding wave propagation coefficients, and boundary condition parameters of a beam subjected to transverse motion.
(2) Description of the Prior Art
By way of example of the state of the art, reference is made to the following papers, which are incorporated herein by reference. Not all of these references may be deemed to be relevant prior art.
D. M. Norris, Jr., and W. C. Young, “Complex Modulus Measurements by Longitudinal Vibration Testing,” Experimental Mechanics, Volume 10, 1970, pp. 93-96.
W. M. Madigosky and G. F. Lee, “Improved Resonance Technique for Materials Characterization,” Journal of the Acoustical Society of America, Volume 73, Number 4, 1983, pp. 1374-1377.
S. L. Garrett, “Resonant Acoustic Determination of Elastic Moduli,” Journal of the Acoustical Society of America, Volume 88, Number 1, 1990, pp. 210-220.
I. Jimeno-Fernandez, H. Uberall, W. M. Madigosky, and R. B. Fiorito, “Resonance Decomposition for the Vibratory Response of a Viscoelastic Rod,” Journal of the Acoustical Society of America, Volume 91, Number 4, Part 1, April 1992, pp. 2030-2033.
G. F. Lee and B. Hartmann, “Material Characterizing System,” U.S. Pat. No. 5,363,701, Nov. 15, 1994.
G. W. Rhodes, A. Migliori, and R. D. Dixon, “Method for Resonant Measurement,” U.S. Pat. No. 5,495,763, Mar. 5, 1996.
R. F. Gibson and E. O. Ayorinde, “Method and Apparatus for Non-Destructive Measurement of Elastic Properties of Structural Materials,” U.S. Pat. No. 5,533,399, Jul. 9, 1996.
B. J. Dobson, “A Straight-Line Technique for Extracting Modal Properties From Frequency Response Data,” Mechanical Systems and Signal Processing, Volume 1, 1987, pp. 29-40.
C. Minas and D. J. Inman, “Matching Finite Element Models to Modal Data,” Journal of Vibration and Acoustics, Volume 112, Number 1, 1990, pp. 84-92,
T. Pritz, “Transfer Function Method for Investigating the Complex Modulus of Acoustic Materials: Rod-Like Specimen,” Journal of Sound and Vibration, Volume 81, 1982, pp. 359-376.
W. M. Madigosky and G. F. Lee, “Instrument for Measuring Dynamic Viscoelastic Properties,” U.S. Pat. No. 4,352,292, Oct. 5, 1982.
W. M. Madigosky and G. F. Lee, “Method for Measuring Material Characteristics,” U.S. Pat. No. 4,418,573, Dec. 6, 1983.
W. Madigosky, “In Situ Dynamic Material Property Measurement System,” U.S. Pat. No. 5,365,457, Nov. 15, 1994.
J. G. McDaniel, P. Dupont, and L. Salvino, “A Wave Approach to Estimating Frequency-Dependent Damping Under Transient Loading” Journal of Sound and Vibration, Volume 231(2), 2000, pp. 433-449.
J. Linjama and T. Lahti, “Measurement of Bending wave reflection and Impedance in a Beam by the Structural Intensity Technique” Journal of Sound and Vibration, Volume 161(2), 1993, pp. 317-331.
L. Koss and D. Karczub, “Euler Beam Bending Wave Solution Predictions of dynamic Strain Using Frequency Response Functions ” Journal of Sound and Vibration, Volume 184(2), 1995, pp. 229-244.
Measuring the flexural properties of beams is important because these parameters significantly contribute to the static and dynamic response of structures. In the past, resonant techniques have been used to identify and measure longitudinal properties. These methods are based on comparing the measured eigenvalues of a structure to predicted eigenvalues from a model of the same structure. The model of the structure must have well-defined (typically closed form) eigenvalues for this method to work. Additionally, resonant techniques only allow measurements at natural frequencies.
Comparison of analytical models to measured frequency response functions is another method used to estimate stiffness and loss parameters of a structure. When the analytical model agrees with one or more frequency response functions, the parameters used to calculate the analytical model are considered accurate. If the analytical model is formulated using a numerical method, a comparison of the model to the data can be difficult due to the dispersion properties of the materials.
Another method to measure stiffness and loss is to deform the material and measure the resistance to the indentation. This method can physically damage the specimen if the deformation causes the sample to enter the plastic region of deformation.